Karl’s Sum: Even famous mathematicians don’t always behave!

There’s a famous story about the mathematician Karl Friedrich Gauss that’s a great one to share with your class.

Even famous mathematicians and their classmates don’t always behave!

One day when students in Karl’s class were really noisy and displaying bad classroom behavior, the teacher got very upset. He gave them a problem to solve. He asked them to find the sum of all the counting numbers from 1 to 100: 1 + 2 + 3 + 4…… + 99 + 100 =

The teacher assumed that the students would be quietly working for the rest of the class period.

But young Karl was done in just a few minutes!

He rearranged all the numbers like this:

(1 + 100) + (2 + 99) + (3 + 98) + . . . . + (50 + 51) = ?

He noticed that every pair of numbers added up to 101. There are 50 pairs of numbers, so the answer is 101 x 50 = 5050.

His teacher was somewhat annoyed that Karl brought up his answer so quickly but when he saw what Karl had done he recognized a mathematician in the making!

After you tell the story and show the pattern at the board. Have students experiment with this themselves by starting with a shorter strand of numbers, such as the sum of the numbers from 1 to 20 or 1 to 30.

See if they can figure out the formula below by observing the patterns.